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Large time behaviour of heat semigroups (and more generally, of positiveselfadjoint semigroups) is studied. Convergence of the semigroup to the groundstate and of averaged logarithms of kernels to the ground state energy is shownin the general framework of positivity improving selfadjoint semigroups. Thisframework includes Laplacians on manifolds, metric graphs and discrete graphs.

Note on basic features of large time behaviour of heat kernels